MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspsstri Structured version   Visualization version   GIF version

Theorem sspsstri 4076
Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
sspsstri ((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Proof of Theorem sspsstri
StepHypRef Expression
1 or32 919 . 2 (((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
2 sspss 4073 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
3 sspss 4073 . . . . 5 (𝐵𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
4 eqcom 2825 . . . . . 6 (𝐵 = 𝐴𝐴 = 𝐵)
54orbi2i 906 . . . . 5 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐵𝐴𝐴 = 𝐵))
63, 5bitri 276 . . . 4 (𝐵𝐴 ↔ (𝐵𝐴𝐴 = 𝐵))
72, 6orbi12i 908 . . 3 ((𝐴𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ (𝐵𝐴𝐴 = 𝐵)))
8 orordir 923 . . 3 (((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ (𝐵𝐴𝐴 = 𝐵)))
97, 8bitr4i 279 . 2 ((𝐴𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵))
10 df-3or 1080 . 2 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
111, 9, 103bitr4i 304 1 ((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wo 841  w3o 1078   = wceq 1528  wss 3933  wpss 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ne 3014  df-in 3940  df-ss 3949  df-pss 3951
This theorem is referenced by:  ordtri3or  6216  sorpss  7443  sorpssi  7444  funpsstri  32905
  Copyright terms: Public domain W3C validator